Wyniki wyszukiwania

1

Recursive expansions

100%

Ash C. J., Knight J. F.

Fundamenta Mathematicae

|

1994

|

tom 145

|

nr 2

153-169

EN

Let A be a recursive structure, and let ψ be a recursive infinitary ${Π}_2$ sentence involving a new relation symbol. The main result of the paper gives syntactical conditions which are necessary and sufficient for every recursive copy of A to have a recursive expansion to a model of ψ, provided A satisfies certain decidability conditions. The decidability conditions involve a notion of rank. The main result is applied to prove some earlier results of Metakides-Nerode and Goncharov. In these applications, the ranks turn out to be low, but there are examples in which the rank takes arbitrary recursive ordinal values.

2

Relatively recursive expansions

100%

Ash C. J., Knight J. F.

Fundamenta Mathematicae

|

1991-1992

|

tom 140

|

nr 2

137-155

EN

In this paper, we consider the following basic question. Let A be an L-structure and let ψ be an infinitary sentence in the language L∪{R}, where R is a new relation symbol. When is it the case that for every B ≅ A, there is a relation R such that (B,R) ⊨ ψ and $R ≤_T D(B)$? We succeed in giving necessary and sufficient conditions in the case where ψ is a "recursive" infinitary $Π_2$ sentence. (A recursive infinitary formula is an infinitary formula with recursive disjunctions and conjunctions.) We consider also some variants of the basic question, in which R is r.e., $Δ_α^0$, or $Σ_α$ instead of recursive relative to D(B).

3

Relatively recursive expansions II

81%

Ash C. J., Knight J. F., Slaman T. A.

Fundamenta Mathematicae

|

1993

|

tom 142

|

nr 2

147-161

EN

In [AK], we asked when a recursive structure A and a sentence φ, with a new relation symbol, have the following property: for each ℬ≅ A there is a relation S such that S is recursive relative to ℬ and ℬ,S)⊨ φ. Here we consider several related properties, in which there is a uniform procedure for determining S from ℬ ≅A, or from ℬ,¯b)≅(A,ā), for some fixed sequence of parameters ā from A; or in which ℬ and S are required to be recursive. We investigate relationships between these properties, showing that for certain kinds of sentences φ, some of these properties do or do not imply others. Many questions are left open.

Ograniczanie wyników

3 Fundamenta Mathematicae

3 Ash C. J.

3 Knight J. F.

1 Slaman T. A.

1 1994

1 1993

1 1992

Recursive expansions

Relatively recursive expansions

Relatively recursive expansions II